Where, when and how to contact me

• Phone: (979)845-2927
• Fax: (979)862-4190
• Email: jzhou@math.tamu.edu
• Office: 641J Blocker Bldg.
• Office Hours: MW 1:00-2:00pm, F 2:00-3:00pm or by appointment.

Education

• Ph.D., Mathematics, Pennsylvania State University, University Park, 1986

Current Teaching Schedule

• Math 251-501, Calculus III, MWF 9:10-10:00am, BLOC 153.
  Homework Assignments First day handout
• Math 251-502, Calculus III, MWF 10:20-11:10am, HELD 109.
  Homework Assignments First day handout

Research Interests

• Multiple Solution Problems, Control Theory, Optimization, Game Theory, Numerical Computation, and Computer Visualization.
• AMS classification numbers: 58E05,35B38,35J65,35J50,65K,65N38
• Selected Publications available.

Current Research Projects

• Computational Theory and Methods for Solving Multiple Solution Problems
• Some interesting findings from this project.
• Download a Matlab code of a Minimax-Newton Method for Solving Semilinear Elliptic PDE for Multiple Solutions (tar file).

Project Descriptions

• Computational Theory and Methods for Solving Multiple Solution Problems.
  Dr. Zhou, is conducting research to develop theory and methods which can be implemented by numerical algorithms for efficiently and stably solving multiple (unstable) solution (eigen) problems in various applications. The problems can be either
  (1) a single PDE (eigen) equation or system,
  (2) variational or non-variational,
  (3) weakly indefinite or strongly indefinite,
  (4) unconstrained or constrained,
  (5) in a Hilbert space or a Banach,
  (6) involving smooth or nonsmooth terms,
  (7) in an interior or exterior domain.
  For each problem, using two-level optimization and game theory, we will
  (1) establish local characterizations of (co-existing) solutions;
  (2) develop numerical algorithms and their codes for finding such multiple solutions; carry out numerical tests on various model problems and make codes available to other researchers;
  (3) do convergence/error analysis of the algorithms; develop techniques to enhance stability, efficiency and convergence rate of the algorithms;
  (4) develop tools that can be numerically carried out for investigating instability and maneuverability of multiple (unstable) solutions;
  (5) carry out numerical study on multiple solution problems in various applications.
  
  Stability is one of the main concerns for system design and control. Traditional critical point theory (Calculus of Variations, Optimization Theory) and numerical methods focus on finding stable solutions. However, in many applications, PERFORMANCE/MANEUVERABILITY is more desirable, especially for system design and control in MISSION CRITICAL SITUATIONS. An unstable solution may have much higher performance/maneuverability than others. One may be willing to take certain risk for pursuing better performance/maneuverability index. It will be interesting to provide a solution balanced performance/maneuverability with its riskiness.
  
  Highly/multiply excited transition states have been observed (can also be laser or electronically induced) in many fields, such as quantum mechanics, condensed matter physics and chemistry, dynamics of biomolecules, nonlinear optics, etc. The life of such states is short and easily to decay to a lower excited state or the ground state by a small perturbation. Such states show variety of configurations and maneuverability, but are very unstable. However, scientists can reach them with new advanced (synchrotronic) technologies and search for NEW applications. It changes people's view about unstable solutions.
  
  When a nonlinear process involves multiple bodies (such as particles, molecules, species, etc.), it leads to a nonlinear system. Comparing to their single equation counterparts, nonlinear systems are much richer in varieties and complexities, and can be classified in many different ways, e.g., cooperative vs noncooperative, definite vs indefinite and different Hamiltonian types. So far, people's understanding of such solutions is still quite limited and analytic solution expressions are too difficult to obtain. Thus development of efficient and reliable numerical methods to solve such problems for multiple solutions becomes more interesting to both theoretical study and applications. Such methods are not yet available elsewhere in the literature, although many papers ( AMS58E05, AMS35B38 and AMS35J10) have proved the existence of such multiple solutions and a large community on saddle computation exists in computational physics/chemistry (Geometry/Transition State/Saddle Optimization).
  
  As a long term research project, Dr. Zhou's research group plans to systematically study computational theory and methods for solving various multiple solution problems. You are welcome to join or collaborate with us.